Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
338 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23×C53 |
| prime radical | C53×#2206 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ C8×Q8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×D8×C53 | C2 | semidirect |
| 2. C53×#2240 | C2 | semidirect |
| 4. C53×#1668 | C2 | semidirect |
| 5. C53×#1364 | C2 | semidirect |
| 6. C53×Q8⋊C16 | C2 | semidirect |
| 7. C53×#1331 | C2 | semidirect |
| 69. D8×C53 | SD16 | semidirect |
| 92. Q8×C53 | D16 | semidirect |
| 115. #12352 | C53 | direct (N × H) |
| 132. Q8×D8 | C2×C53 | semidirect |
| 133. #2240 | C2×C53 | semidirect |
| 135. #1668 | C2×C53 | semidirect |
| 136. #1364 | C2×C53 | semidirect |
| 137. Q8⋊C16 | C2×C53 | semidirect |
| 138. #1331 | C2×C53 | semidirect |
| 161. C53 | #12352 | direct (N × H) |
| 206. D8 | SD16×C53 | semidirect |
| 229. Q8 | D16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×D8×C53 | 2 |
| C53×#2240 | 2 |
| C53×#2206 | 2 |
| C53×#1668 | 2 |
| C53×#1364 | 2 |
| C53×Q8⋊C16 | 2 |
| C53×#1331 | 2 |
| #12352 | 53 |